Given any △ABC ([color=#1551b5]blue[/color]), you can construct three external equilateral triangles ([color=#0a971e]green[/color]).[br][br]Napoleon's Theorem states that if you connect the centroids of the these triangles, the resulting △XYZ ([color=#c51414]red[/color]) will also be an equilateral triangle.[br][br]On the worksheet below, manipulate points A, B and C to see how changes △ABC affect the proportions of △XYZ. Use the guiding questions below to explore the properties of Napoleon's Theorem.[br][br]The following notes are essential for proving the theorem. [br][list=1][br][*]NL is rotated 30° clockwise around point B to yield N'L'. [br][*]ML is rotated 30° counterclockwise around point C to yield M'L''.[br][*]Both N'L' and M'L'' dilated by a k = 2*cos(30°) from points B and C respectively result in AP.[br][*]Because these three lines can be obtained through dilations, we know they are all parallel.[br][*]Further, because both are a rotation of 30° of the original segments, we can show that m∠NLM = 60°.[br][*]Given that △NLM is an isosceles and ∠NLM is NOT one of the base angles, we can prove that △NLM is equilateral.[br][/list]

[list=1][br][*]Is the AREA of △NLM dependent on the AREA of △ABC? To explore this, see what happens when points A, B and C are co-linear.[br][*]The statement above says that the three green triangles should be EXTERNAL to △ABC. Does it appear to be necessary that they are external?[br][*]What happens to the ratios as you manipulate points A, B and C?[br][/list]